Question

If f(x)=x2−14x+49 is true for x≥7, find f⁻¹ (x).

a) f⁻¹1(x)=(x−7)²
b) f⁻¹ (x)=x+7
c) f⁻¹ (x)=√x+7
d) f⁻¹1(x)=x−7

Answer 1

To find the inverse function f⁻¹(x) of f(x) = x² − 14x + 49 for x≥7, swap x and f(x) and solve for the new variable. The inverse function is f⁻¹(x) = ±√x + 7.

Step-by-step explanation:

To find the inverse function f⁻¹(x) when given f(x) = x² − 14x + 49 for x≥7, we can use the method of switching the roles of x and f(x) and solve for the new variable, which in this case is f⁻¹(x). Let's proceed with the steps:

1. Exchange x and f(x): x = y² − 14y + 49
2. Now solve for y, which will be the inverse function: y² − 14y + 49 - x = 0
3. Completing the square we have (y - 7)² = x
4. Taking the square root of both sides we get y - 7 = ±√x
5. Finally, adding 7 to both sides we have f⁻¹(x) = ±√x + 7

Therefore, the inverse function of f(x) = x² − 14x + 49 for x≥7 is f⁻¹(x) = ±√x + 7.

For the function f(x) = x2 - 14x + 49, which applies for x ≥ 7, we are looking to find the inverse function, f−1(x). To find the inverse, we need to perform algebraic manipulations that essentially 'undo' what the original function does. Given that the function is a quadratic and the coefficient before the x2 term is 1, we realize the original function is a perfect square trinomial, which factors into (x - 7)2. To find the inverse, we set y = (x - 7)2 and then swap x and y. This gives us x = (y - 7)2, and to solve for y, we take the square root of both sides, considering the domain x ≥ 7, and isolate y, yielding y = √x + 7. From the options provided, the correct inverse function is f−1(x) = √x + 7, or option c).